Update The plane wave state for a free particle given by $\psi(x,t)=A\exp[i(kx-\omega t)]$ is completely delocalized in space and time. Therefore, the wavefunction is present everywhere in space at all times. We can, however, 'normalize' this wavefunction, pretending that it is confined in a box of length $L$. In this case, we find, $$\psi(x,t)=\frac{1}{\sqrt{L}}\exp[i(k_nx-\omega t)]$$ with quantized wavenumber $k\rightarrow k_n$. The position probability density has a spatial profile $$\rho(x)=\psi^*\psi=1/L$$ which is independent of time (like any other stationary state). Moreover, the probability of finding the particle is same everywhere at all times. But if we calculate the probability current (flow) it is nonzero: $$j(x)=-\frac{\hbar k}{m}\frac{1}{L}\neq 0.$$ For example, $J_\phi$, the azimuthal component of the current is nonzero in Hydrogen atom for the stationary states.
Therefore, although the spatial profile of position probability density does not change with time, there is a propagation!
How do we interpret this? Should it be interpreted that at any point $x$ the influx of probability is equal to the outflux such that the probability amplitude at any point remains constant in time?
